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Nonlocal Diffusion Models with Consistent Local and Fractional Limits

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A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 165))

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Abstract

For some spatially nonlocal diffusion models with a finite range of nonlocal interactions measured by a positive parameter \(\delta \), we review their formulation defined on a bounded domain subject to various conditions that correspond to some inhomogeneous data. We consider their consistency to similar inhomogeneous boundary value problems of classical partial differential equation (PDE) models as the nonlocal interaction kernel gets localized in the local \(\delta \to 0\) limit, and at the same time, for rescaled fractional-type kernels, to corresponding inhomogeneous nonlocal boundary value problems of fractional equations in the global \(\delta \to \infty \) limit. Such discussions help to delineate issues related to nonlocal problems defined on a bounded domain with inhomogeneous data.

The research of Q. Du was supported in part by NSF grant DMS-2012562 and DMS-1937254. The research of X. Tian was supported in part by NSF grant DMS-2111608. The research of Z. Zhou is partially supported by Hong Kong Research Grants Council (No. 15303122) and an internal grant of Hong Kong Polytechnic University (Project ID: P0031041, Programme: ZZKS)

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Authors and Affiliations

  1. Department of Applied Physics and Applied Mathematics, and Data Science Institute, Columbia University, New York, NY, USA

    Qiang Du

  2. Department of Mathematics, University of California, San Diego, CA, USA

    Xiaochuan Tian

  3. Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China

    Zhi Zhou

Authors
  1. Qiang Du
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  2. Xiaochuan Tian
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  3. Zhi Zhou
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Correspondence to Qiang Du .

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Editors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, TN, USA

    Tadele Mengesha

  2. Department of Mathematics, University of Tennessee, Knoxville, TN, USA

    Abner J. Salgado

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Du, Q., Tian, X., Zhou, Z. (2023). Nonlocal Diffusion Models with Consistent Local and Fractional Limits. In: Mengesha, T., Salgado, A.J. (eds) A³N²M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models. The IMA Volumes in Mathematics and its Applications, vol 165. Springer, Cham. https://doi.org/10.1007/978-3-031-34089-5_5

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